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184_notes:b_current [2017/10/04 18:44] – dmcpadden | 184_notes:b_current [2021/06/16 18:53] – [Magnetic field from Many Charges] bartonmo | ||
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+ | Sections 17.2 and 17.6-17.8 in Matter and Interactions (4th edition) | ||
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+ | [[184_notes: | ||
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===== Currents Make Magnetic Fields ===== | ===== Currents Make Magnetic Fields ===== | ||
- | The next source of magnetic fields that we are going to consider is currents (either comprised of electrons or some other charged particle). | + | Now that we have talked about a single moving charge and permanent magnets, the next source of magnetic fields that we are going to consider is currents (either comprised of electrons or some other charged particle). |
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+ | {{youtube> | ||
==== Magnetic field from Many Charges ==== | ==== Magnetic field from Many Charges ==== | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
- | If we consider a straight wire with a steady current, there would be many moving charges everywhere in wire that would all contribute to the magnetic field outside of the wire. If we take a " | + | If we consider a straight wire with a steady current, there would be many moving charges everywhere in wire that would all contribute to the magnetic field outside of the wire. If we take a " |
$$\vec{B}_{tot}=\frac{\mu_0}{4 \pi}\frac{q_1\vec{v}\times \hat{r_1}}{r_1^2}+\frac{\mu_0}{4 \pi}\frac{q_2\vec{v}\times \hat{r_2}}{r_2^2}+\frac{\mu_0}{4 \pi}\frac{q_3\vec{v}\times \hat{r_3}}{r_3^2}+...=\Sigma_i \frac{\mu_0}{4 \pi}\frac{q_i\vec{v}\times \hat{r_i}}{r_i^2}$$ | $$\vec{B}_{tot}=\frac{\mu_0}{4 \pi}\frac{q_1\vec{v}\times \hat{r_1}}{r_1^2}+\frac{\mu_0}{4 \pi}\frac{q_2\vec{v}\times \hat{r_2}}{r_2^2}+\frac{\mu_0}{4 \pi}\frac{q_3\vec{v}\times \hat{r_3}}{r_3^2}+...=\Sigma_i \frac{\mu_0}{4 \pi}\frac{q_i\vec{v}\times \hat{r_i}}{r_i^2}$$ | ||
Since we have many small charges that we are adding the field contributions from, we can turn the summation into an integral and the individual charges $q_i$ into $dq$: | Since we have many small charges that we are adding the field contributions from, we can turn the summation into an integral and the individual charges $q_i$ into $dq$: | ||
$$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{dq \cdot \vec{v}\times \hat{r}}{r^2}$$ | $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{dq \cdot \vec{v}\times \hat{r}}{r^2}$$ | ||
- | Again this equation just says that we are going to add together the magnetic field contributions at a point from every charge (dq) that is moving in the wire. | + | Again this equation just says that we are going to add together the magnetic field contributions at a point from every charge ($dq$) that is moving in the wire. |
- | Now we can rewrite the velocity in terms of the differential length and time: $\vec{v}=\frac{d\vec{l}}{dt}$. In other words, velocity is simply the change in displacement over a change in time. | + | Now we can rewrite the velocity in terms of the differential length and time: $\vec{v}=\frac{d\vec{l}}{dt}$. In other words, velocity is simply the change in displacement |
$$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{dq \cdot \frac{d\vec{l}}{dt}\times \hat{r}}{r^2}$$ | $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{dq \cdot \frac{d\vec{l}}{dt}\times \hat{r}}{r^2}$$ | ||
- | Since dt represents a small amount of time, dl represents a small amount of length, and dq represents a small amount of charge, we will treat these as independent and rewrite (much to the chagrin of our mathematician friends): | + | Since $dt$ represents a small amount of time, $dl$ represents a small amount of length, and $dq$ represents a small amount of charge, we will treat these as independent and rewrite (much to the chagrin of our mathematician friends): |
$$dq \cdot \frac{d\vec{l}}{dt} = \frac{dq \cdot d\vec{l}}{dt}=\frac{dq}{dt}\cdot d\vec{l}$$ | $$dq \cdot \frac{d\vec{l}}{dt} = \frac{dq \cdot d\vec{l}}{dt}=\frac{dq}{dt}\cdot d\vec{l}$$ | ||
So our magnetic field equation then becomes: | So our magnetic field equation then becomes: | ||
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We can now use the definition of [[184_notes: | We can now use the definition of [[184_notes: | ||
$$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | ||
- | **Note that $I$ here is the //conventional// current**, not the electron current. Otherwise many of the pieces of this equation would be what you expected: | + | **Note that $I$ here is the conventional current, not the electron current**. Otherwise many of the pieces of this equation would be what you expected: |
- | {{ 184_notes: | + | [{{ 184_notes: |
*The constant is the same as before (we haven' | *The constant is the same as before (we haven' | ||
*The current $I$ tells you about the amount of charge per second flowing through the wire. This is a scalar number with units of Amps where $A=\frac{C}{s}$. | *The current $I$ tells you about the amount of charge per second flowing through the wire. This is a scalar number with units of Amps where $A=\frac{C}{s}$. | ||
*The length $d\vec{l}$ is now what we are integrating over - so we want to add up all the little bits of the wire that have current flowing through them. Since $d\vec{l}$ originally came from the velocity vector, $d\vec{l}$ should point in the same direction that the charges are moving in. | *The length $d\vec{l}$ is now what we are integrating over - so we want to add up all the little bits of the wire that have current flowing through them. Since $d\vec{l}$ originally came from the velocity vector, $d\vec{l}$ should point in the same direction that the charges are moving in. | ||
- | * The $\vec{r}$ (and $r$/$\hat{r}$) is then the separation vector that point between the $d\vec{l}$ (the source) and the observation location. | + | * The $\vec{r}$ (and relatedly |
* The cross product between $d\vec{l}$ and $\hat{r}$ will still give us a direction for the magnetic field that is perpendicular to the separation vector and the direction that the charges move. | * The cross product between $d\vec{l}$ and $\hat{r}$ will still give us a direction for the magnetic field that is perpendicular to the separation vector and the direction that the charges move. | ||
- | We will go into detail about how to put the pieces of this equation together in an example; however, it is important to realize that this equation doesn' | + | We will go into detail about how to put the pieces of this equation together in an example; however, it is important to realize that this equation doesn' |
==== Magnetic Field from a Very Long Wire ==== | ==== Magnetic Field from a Very Long Wire ==== | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
Let's look at a particular example of finding the magnetic field a distance $s$ away from a very long wire with some // | Let's look at a particular example of finding the magnetic field a distance $s$ away from a very long wire with some // | ||
$$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | ||
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==== Examples ==== | ==== Examples ==== | ||
- | Magnetic | + | [[: |